Solve 0=3x^3-5x+2

$$0=3 { x }^{ 3 } -5x+2$$

Solve for x

$$x=-\frac{\sqrt{33}}{6}-\frac{1}{2}\approx -1.457427108$$

$$x=1$$

$$x=\frac{\sqrt{33}}{6}-\frac{1}{2}\approx 0.457427108$$

Solution Steps

Swap sides so that all variable terms are on the left hand side.

$$3x^{3}-5x+2=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 2 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{2}{3},±2,±\frac{1}{3},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{3}-5x+2 by x-1 to get 3x^{2}+3x-2. Solve the equation where the result equals to 0.

$$3x^{2}+3x-2=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 3 for a, 3 for b, and -2 for c in the quadratic formula.

$$x=\frac{-3±\sqrt{3^{2}-4\times 3\left(-2\right)}}{2\times 3}$$

Do the calculations.

$$x=\frac{-3±\sqrt{33}}{6}$$

Solve the equation 3x^{2}+3x-2=0 when ± is plus and when ± is minus.

$$x=-\frac{\sqrt{33}}{6}-\frac{1}{2} x=\frac{\sqrt{33}}{6}-\frac{1}{2}$$

List all found solutions.

$$x=1 x=-\frac{\sqrt{33}}{6}-\frac{1}{2} x=\frac{\sqrt{33}}{6}-\frac{1}{2}$$